Intereting Posts

Cardinality of the intersection of 2 power sets
How to find this limit: $A=\lim_{n\to \infty}\sqrt{1+\sqrt{\frac{1}{2}+\sqrt{\frac{1}{3}+\cdots+\sqrt{\frac{1}{n}}}}}$
counting monotonically increasing functions
bijection between prime ideals of $R_p$ and prime ideals of $R$ contained in $P$
Prove that $x\sqrt{1-x^2} \leq \sin x \leq x$
Real projective space $\mathbb{R}P^1$ is diffeomorphic to $S^1$?
Exponential Function of Quaternion – Derivation
Show that, for all $n > 1: \log \frac{2n + 1}{n} < \frac1n + \frac{1}{n + 1} + \cdots + \frac{1}{2n} < \log \frac{2n}{n – 1}$
If $g:V \rightarrow V$ is an injective linear transformation. Prove if $V$ is finite dimensional then $g$ is surjective.
If $f(x) = h(x)g(x)$, is $h$ differentiable if $f$ and $g$ are?
Unitary Farey Sequence Matrices
Reference for a proof of the Hahn-Mazurkiewicz theorem
Why are properties lost in the Cayley–Dickson construction?
Calculate the closed form of $\frac{\sqrt{5}}{\sqrt{3}}\cdot \frac{\sqrt{9}}{\sqrt{7}}\cdot \frac{\sqrt{13}}{\sqrt{11}}\cdot …$
Geometry Construction Problems

Suppose I embed a manifold-with-boundary $M$ in some $\mathbb{R}^n$. Are there conditions (necessary, sufficient, or both) that can help determine when the topological boundary of $M$ is equal to the manifold boundary?

By “topological boundary,” I’m referring to $\text{Bd } M$, which is the closure minus the interior (relative to $\mathbb{R}^n$).

By “manifold boundary,” I mean the boundary $\partial M$ that is specified in the definition of “manifold-with-boundary.”

- Embedding manifolds of constant curvature in manifolds of other curvatures
- Prerequisite for Petersen's Riemannian Geometry
- Killing Field on a Riemannian Manifold
- Examples of 2-dimensional foliations of a 4-sphere.
- Divergence of vector field on manifold
- Are there different ways to embed surface with nonvanishing curvature in a higher-dimensional Euclidean space?

- First proof of Poincaré Lemma
- What is a conormal vector to a domain intuitively?
- Is a twisted de Rham cohomology always the same as the untwisted one?
- The Gaussian and Mean Curvatures of a Parallel Surface
- Why every map $f : S^n \to T^n (n>1)$ has topological degree zero?
- What does $dx$ mean in differential form?
- Why does the tangent vector measure the rate of change of the angle which neighboring tangent make with the tangent?
- The Hodge $*$-operator and the wedge product
- Why maximal atlas
- Riemannian Manifolds with $n(n+1)/2$ dimensional symmetry group

If you embed a smooth $m$-manifold $M$ smoothly in $\mathbb{R}^n$

then the “topological boundary” of $M$ is the closure of $M$. As

locally each point of $M$ has a neighbourhood in $\mathbb{R}^n$

where $M$ looks like $\mathbb{R}^m$, then no point of $M$ is interior.

When $m=n$ and $M$ is compact then, yes (at least in the smooth case)

the topological and manifold boundaries coincide.

I expect the above hold for topological embeddings but won’t swear

to it; they need not be locally flat (see nasties like the Alexander

horned sphere),

- What is the mathematical notation for a random number in a certain range?
- Prove that the expression cannot be a power of 2
- Is this integral improper? If yes – why?
- Why is cross product only defined in 3 and 7 dimensions?
- Proof that $\inf A = \sup B$
- Does $A$ a UFD imply that $A$ is also a UFD?
- Trying to evaluate $\prod_{k=1}^{n-1}(1-e^{2k\pi i/n})$ for my complex analysis homework
- Distributive nearring
- A minimization problem in function fitting setup
- Need help with intro question about complex polynomials
- The kernel and range of the powers of a self-adjoint operator
- Find $\Big\{ (a,b)\ \Big|\ \big|a\big|+\big|b\big|\ge 2/\sqrt{3}\ \text{ and }\forall x \in\mathbb{R}\ \big|a\sin x + b\sin 2x\big|\le 1\Big\}$
- Basic question regarding a finitely generated graded $A$-algebra
- Independence of Roots
- Axis of Symmetry for a General Parabola